An analysis of the nonlinear equation of motion of a vibrating membrane in the space of BV functions

Dedicated to Professor Kiyoshi Mochizuki on his sixtieth birthday

Abstract

In this article the nonlinear equation of motion of vibrating membrane u_tt-div{√{1+|∇ u|²}^-1∇ u}=0 is discussed in the space of functions having bounded variation. Approximate solutions are constructed in Rothe's method. It is proved that a subsequence of them converges to a function u and that, if u satisfies the energy conservation law, then it is a weak solution in the space of functions having bounded variation. The main tool is varifold convergence.